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SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 61 , Issue 3 : 039811(2018) https://doi.org/10.1007/s11433-017-9111-4

Diagnosing holographic type dark energy models with the Statefinder hierarchy, composite null diagnostic and $w$-$w'$ pair

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  • ReceivedAug 30, 2017
  • AcceptedSep 28, 2017
  • PublishedDec 15, 2017
PACS numbers

Abstract


Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant No. 11405024), and the Fundamental Research Funds for the Central Universities (Grant No. 16lgpy50). We thank LanJun Zhou and Nan Li for valuable suggestions and technological supports.


References

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  • Figure 1

    (Color online) Evolutions of $S^{(1)}_3$ versus redshift $z$ for the HDE model (a1, a2) and the $\varLambda$HDE model (b1, b2), with varying $\varOmega_{{\rm~de}0}$ (a1, b1) and varying $C$ (a2, b2). Note that the result of the $\varLambda$CDM is also plotted as a solid horizontal line for comparison.

  • Figure 2

    (Color online) Evolutions of $S^{(1)}_4$ versus redshift $z$ for the HDE model (a1, a2) and $\varLambda$HDE model (b1, b2), with varying $\varOmega_{{\rm~de}0}$ (a1, b1) and varying $C$ (a2, b2). Note that the result of the $\varLambda$CDM model is also plotted as a solid horizontal line for comparison.

  • Figure 3

    (Color online) Evolutions of $\epsilon$ versus redshift $z$ for the HDE model (a1, a2) and $\varLambda$HDE model (b1, b2), with varying $\varOmega_{{\rm~de}0}$ (a1, b1) and varying $C$ (a2, b2). Note that the result of the $\varLambda$CDM model is also plotted as a solid horizontal line for comparison.

  • Figure 4

    (Color online) The evolutionary trajectories of the CND pair $S_3^{(1)},~\epsilon$for the HDE model (a1, a2) and the $\varLambda$HDE model (b1, b2), with varying $\varOmega_{{\rm~de}0}$ (a1, b1) and varying $C$ (a2, b2). The current values of $S_3^{(1)},~\epsilon$for the HDE and $\varLambda$HDE are marked by solid dots. For the $\varLambda$CDM model, $\{S^{(1)}_3,~\epsilon\}=\{1,1\}$ is marked by hexagram for comparison. Note that the arrows indicate the time directions of cosmic evolution ($z~\to~0$).

  • Figure 5

    (Color online) The evolutionary trajectories of the CND pair $S_4^{(1)},~\epsilon$for the HDE (a1, a2) and $\varLambda$HDE (b1, b2), with varying $\varOmega_{{\rm~de}0}$ (a1, b1) and varying $C$ (a2, b2). The current values of $S_4^{(1)},~\epsilon$for the HDE model and the $\varLambda$HDE model are marked by solid dots. For the $\varLambda$CDM model, $\{S^{(1)}_4,~\epsilon\}=\{1,1\}$ is marked by hexagram for comparison. Note that the arrows indicate the time directions of cosmic evolution ($z~\to~0$).

  • Figure 6

    (Color online) Comparisons of the evolutionary trajectories of $\{S^{(1)}_3~$(a), $\epsilon\}$ and $\{S^{(1)}_4,~\epsilon\}$ for the HDE, $\varLambda$HDE, NADE, and RDE models in the $S^{(1)}_3$–$\epsilon$ plane. The current values of $\{S^{(1)}_3,~\epsilon\}$ and $\{S^{(1)}_4,~\epsilon\}$ for the holographic type DE models are marked by the round dots. $\{S^{(1)}_3,~\epsilon\}=\{1,1\}$ and $\{S^{(1)}_4,~\epsilon\}=\{1,1\}$ for the $\varLambda$CDM model is also shown as hexagram for comparison. The arrows indicate the evolution directions of the models.

  • Table 1   The present values of the Statefinder hierarchy pairs $S^{(1)}_{3t_{0}}$, $S^{(1)}_{4t_{0}}$, and their maximum differences, $\Delta~S^{(1)}_{3t_{0}}$ and $\Delta~S^{(1)}_{4t_{0}}$, for HDE and $\varLambda$HDE with varying $\varOmega_{{\rm~de}0}$, where $\Delta~S^{(1)}_{3t_{0}}=S^{(1)}_{3t_{0}}({\rm~max})-S^{(1)}_{3t_{0}}(\rm{min})$ and $\Delta~S^{(1)}_{4t_{0}}=S^{(1)}_{4t_{0}}({\rm~max})-S^{(1)}_{4t_{0}}(\rm{min})$ for each model
    HDE $(C=0.6)$ $\varLambda$HDE $(\varOmega_{\varLambda~0}=0.4,~C=0.6)$
    $\varOmega_{{\rm~de}0}$ $0.5$ $0.6$ $0.7$ $0.8$ $0.5$ $0.6$ $0.7$ $0.8$
    $S^{(1)}_{3t_{0}}$ $1.79$ $2.18$ $2.60$ $3.04$ $0.92$ $0.93$ $1.03$ $1.21$
    $S^{(1)}_{4t_{0}}$ $3.27$ $4.17$ $5.20$ $6.40$ $1.04$ $1.04$ $1.16$ $1.44$
    $\Delta~S^{(1)}_{3t_{0}}$ $1.25$ $0.29$
    $\Delta~S^{(1)}_{4t_{0}}$ $3.13$ $0.40$
  • Table 2   The present values of the Statefinders hierarchy pairs $S^{(1)}_{3t_{0}}$, $S^{(1)}_{4t_{0}}$, and their maximum differences, $\Delta~S^{(1)}_{3t_{0}}$ and $\Delta~S^{(1)}_{4t_{0}}$, for HDE and $\varLambda$HDE with varying $C$, where $\Delta~S^{(1)}_{3t_{0}}=S^{(1)}_{3t_{0}}({\rm~max})-S^{(1)}_{3t_{0}}(\rm{min})$ and $\Delta~S^{(1)}_{4t_{0}}=S^{(1)}_{4t_{0}}({\rm~max})-S^{(1)}_{4t_{0}}(\rm{min})$ for each model
    HDE $(\varOmega_{{\rm~de}0}=0.7)$ $\varLambda$HDE $(\varOmega_{\varLambda~0}=0.4,~\varOmega_{\rm~hde0}=0.3)$
    C $0.2$ $0.4$ $0.6$ $0.8$ $0.2$ $0.4$ $0.6$ $0.8$
    $S^{(1)}_{3t_{0}}$ $25.99$ $6.10$ $2.60$ $1.44$ $5.75$ $1.71$ $1.03$ $0.82$
    $S^{(1)}_{4t_{0}}$ $278.19$ $24.69$ $5.20$ $1.91$ $38.55$ $3.28$ $1.16$ $0.98$
    $\Delta~S^{(1)}_{3t_{0}}$ $24.55$ $4.93$
    $\Delta~S^{(1)}_{4t_{0}}$ $276.28$ $37.57$
qqqq

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